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<style|generic>

<\body>
  <doc-data|<doc-title|Fourier Transform>>

  Fourier Transform can have 4 types of transforms :

  1. Continuous Time <math|\<rightarrow\>> Continuous Frequency

  <\equation*>
    X<around*|(|f|)>=<big|int><rsup|\<infty\>><rsub|-\<infty\>>x<around*|(|t|)>e<rsup|-j2\<pi\>f
    t >d t
  </equation*>

  2. Continuous Time <math|\<rightarrow\>> Discrete Frequency

  3. Discrete time <math|\<rightarrow\>> Discrete Frequency

  4. Discrete time <math|\<rightarrow\>> Continuous Frequency

  \;

  \;

  Fourier Transform can be written using frequency <math|f> or
  <math|\<omega\>>

  <\equation*>
    X<around*|(|f|)>=<big|int><rsup|\<infty\>><rsub|-\<infty\>>x<around*|(|t|)>e<rsup|-j2\<pi\>f
    t> d t
  </equation*>

  <\equation*>
    X<around*|(|\<omega\>|)>=<big|int><rsup|\<infty\>><rsub|-\<infty\>>x<around*|(|t|)>e<rsup|-j\<omega\>t>
    d\<omega\>
  </equation*>

  The inverse Fourier Transform can be written similarly

  <\equation*>
    x<around*|(|t|)>=<big|int><rsup|\<infty\>><rsub|-\<infty\>>X<around*|(|f|)>e<rsup|j2\<pi\>f
    t> d f
  </equation*>

  <\equation*>
    x<around*|(|t|)>=<frac|1|2\<pi\>><big|int><rsup|\<infty\>><rsub|-\<infty\>>X<around*|(|\<omega\>|)>e<rsup|j
    \<omega\>t>d\<omega\>
  </equation*>

  The reason why there is <math|<frac|1|2\<pi\>>> is because
  <math|\<omega\>=2\<pi\> f> , therefore <math|d\<omega\>=2\<pi\> >

  <\equation*>
    \;
  </equation*>
</body>

<\initial>
  <\collection>
    <associate|language|american>
    <associate|page-type|letter>
  </collection>
</initial>